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Definition df-dmd 28306
Description: Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 28324 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-dmd 𝑀* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))))}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-dmd
StepHypRef Expression
1 cdmd 26990 . 2 class 𝑀*
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 cch 26952 . . . . . 6 class C
53, 4wcel 1938 . . . . 5 wff 𝑥C
6 vy . . . . . . 7 setvar 𝑦
76cv 1473 . . . . . 6 class 𝑦
87, 4wcel 1938 . . . . 5 wff 𝑦C
95, 8wa 382 . . . 4 wff (𝑥C𝑦C )
10 vz . . . . . . . 8 setvar 𝑧
1110cv 1473 . . . . . . 7 class 𝑧
127, 11wss 3444 . . . . . 6 wff 𝑦𝑧
1311, 3cin 3443 . . . . . . . 8 class (𝑧𝑥)
14 chj 26956 . . . . . . . 8 class
1513, 7, 14co 6431 . . . . . . 7 class ((𝑧𝑥) ∨ 𝑦)
163, 7, 14co 6431 . . . . . . . 8 class (𝑥 𝑦)
1711, 16cin 3443 . . . . . . 7 class (𝑧 ∩ (𝑥 𝑦))
1815, 17wceq 1474 . . . . . 6 wff ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))
1912, 18wi 4 . . . . 5 wff (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦)))
2019, 10, 4wral 2800 . . . 4 wff 𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦)))
219, 20wa 382 . . 3 wff ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))))
2221, 2, 6copab 4540 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))))}
231, 22wceq 1474 1 wff 𝑀* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ ∀𝑧C (𝑦𝑧 → ((𝑧𝑥) ∨ 𝑦) = (𝑧 ∩ (𝑥 𝑦))))}
Colors of variables: wff setvar class
This definition is referenced by:  dmdbr  28324
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